Massive MIMO-Based Distributed Signal Detection in Multi-Antenna Wireless Sensor Networks ^†

This paper considers a massive MIMO-based distributed detection in multi-antenna wireless sensor networks, as shown in Figure 1, over a “virtual” multiple-input multiple-output (MIMO) channel between N multi-antenna sensors (K antennas per sensor) and FC configured with M antennas. The sensors will perceive observations of information within their coverage. The local decision on the presence of a signal is sent to the fusion center for global judgment. This scenario can be modeled as a distributed binary hypothesis testing problem [22], where $H_0$ and $H_1$ indicate the presence and absence of the signal $\theta$, respectively. The sensor measurement model under the two hypotheses is [35] where $v_i\sim \mathcal{C}\mathcal{N}(0,{\sigma }_{v,i}^2)$ is the measurement noise of the i-th sensor, the signal of interest $\theta$ is modeled as a zero-mean circular complex Gaussian variable with variance ${\sigma }_{\theta }^2$, a distribution we denote by $\theta \sim \mathcal{C}\mathcal{N}(0,{\sigma }_{\theta }^2)$.

The k-th antenna of the i-th sensor amplifies its measurement $s_i$ with a gain $g_{ik}$ and forward it to the M-antenna FC through a coherent multiple access channel. The received signal at the m-th antenna of the FC is where ${\mathrm{g}}_{ik}$ in Equation (2) is the k-th antenna gain of the i-th sensor, and $1\le i\le N,1\le k\le K,1\le m\le M,$ ${\mathrm{h}}_{ik,m}$ indicates the channel gain between the k-th antenna of the i-th sensor and the m-th antenna of the FC, and the wireless fading channel is modeled as where $d_i$ is the distance between the i-th sensor and the FC. Since the different antennas of the same sensor are located similarly, we consider that the distance between the antennas on the same sensor is relatively close, which is far less than the distance between the sensor and the FC, in order to simplify the processing, the channel gain between different antennas of the same sensor and the FC is approximately the same, so ${\mathrm{h}}_{ik,\mathrm{m}}\approx h_{i,m}$, $\alpha$ is the path loss index factor, ${\tilde{h}}_{i,m}\sim \mathcal{C}\mathcal{N}(0,I_M)$ is a complex Gaussian vector.

Substituting Equation (2) into Equation (1),

To facilitate derivation and memory, the symbol $\mathit{g}=[g_{11},\dots ,g_{ik}{]}^T$ (dimension: $i\ast k$ row, 1 column) is introduced to indicate the antenna gain of each antenna of the sensor. In addition, $\mathit{g}=\sqrt{\mathit{P}}$, $\mathit{P}=[p_{11},\dots ,p_{ik}]$ (dimension: 1 row, $i\ast k$ column) is the reported power of different antennas of i sensors, and $\mathit{D}$$=diag[g_{11},\dots ,g_{ik}]$ (dimension: $i\ast k$ row, $i\ast k$ column), $\mathit{v}={[v_{11},\dots ,v_{ik}]}^T$ (dimension: $i\ast k$ row, 1 column) represents the noise on each antenna of the sensor, and $\mathit{H}$$=[{\mathit{h}}_{\mathbf{11}},\dots ,{\mathit{h}}_{\mathit{ik}}]$ (dimension: m row, $i\ast k$ column) is the channel gain matrix, where ${\mathrm{h}}_{\mathit{i}\mathit{k}}={[h_{ik,1},\dots ,h_{ik,m}]}^T$ (dimension: m row, 1 column). In distributed multi-antenna wireless sensor networks based on massive MIMO, it is assumed that the matrix $\mathit{H}$ corresponds to the different column vectors for different sensors and is independent of each other, $\mathit{n}={[n_1,\dots ,n_M]}^T$ (dimension: m row, 1 column) $\sim \mathcal{C}\mathcal{N}(0,{\sigma }_n^2{\mathrm{I}}_M)$ is the noise at the fusion center.

Therefore, the above symbols indicate that the signal received at the FC is [36]

In this section, we determine the presence of the signal based on the received signal $\mathit{y}$. We distinguish between $\mathrm{D}={\mathcal{H}}_0$ and $\mathrm{D}={\mathcal{H}}_1$ using a likelihood ratio test (LRT) called the best fusion rule. FC maximizes the probability of detection by a Neyman–Pearson (NP) criterion for a given false alarm probability. The LRT detector works at the threshold $\gamma$ as follows:

In Equation (6), $\mathrm{p}\left(y\right|{\mathcal{H}}_1)$ and $\mathrm{p}\left(y\right|{\mathcal{H}}_0)$ are the probability density function (PDF) of the receiving matrix is under the alternative hypothesis ${\mathcal{H}}_1$ and the null hypothesis ${\mathcal{H}}_0$, respectively. According to the Gaussian distribution probability density function $f\left(x\right)=\frac{1}{\sqrt{2\pi {\sigma }^2}}exp(-\frac{{(x-\mu )}^2}{2{\sigma }^2})$, we have [37] where ${\mathit{C}}_w=\mathit{H}\mathit{D}\mathit{V}{\mathit{D}}^H{\mathit{H}}^H+{\sigma }_n^2{\mathit{I}}_M$ is the covariance of $\mathit{y}$ under ${\mathcal{H}}_0$, and ${\mathit{C}}_s={\sigma }_{\theta }^2\mathit{H}\mathit{g}{\mathit{g}}^H{\mathit{H}}^H$, so ${\mathit{C}}_w+{\mathit{C}}_s$ is the covariance of $\mathit{y}$ under ${\mathcal{H}}_1$, where $\mathit{V}=diag[{\sigma }_{v,11}^2,\dots ,{\sigma }_{v,ik}^2]$ is the measurement noise variance diagonal matrix of multi-antenna sensors.

To get the specific expression of the detector, substituting Equation (7) into Equation (6), then and we have

Furthermore, we can get where the symbol $\widehat{\gamma }=(1+{\sigma }_{\theta }^{2}f\left(\mathit{g}\right)\left)ln\right[\gamma (1+{\sigma }_{\theta }^{2}f\left(\mathit{g}\right)\left)\right]$ and $f\left(\mathit{g}\right)={\mathit{g}}^H{\mathit{H}}^H{\mathit{C}}_w^{-1}\mathit{H}\mathit{g}$.

The detection probability $P_{D,i}$ and the false alarm probability $P_{FA,i}$ of the i-th sensor are defined as

Based on the obtained LRT detector expression, we evaluate the FC’s detection performance by defining and calculating the detection probability ${\mathrm{P}}_D$ and the false alarm probability $P_{FA}$. Specifically, the detection probability that a wireless signal exists and is detected, is defined as [18]

We can rewrite Equation (13) as where $\tilde{\mathit{y}}=({\mathit{C}}_w+{\mathit{C}}_s{)}^{-\frac{1}{2}}\mathit{y}$ and $\mathit{W}=({\mathit{C}}_w+{\mathit{C}}_s{)}^{\frac{1}{2}}{\mathit{C}}_w^{-1}\mathit{Hg}{\mathit{g}}^H{\mathit{H}}^H{\mathit{C}}_w^{-1}({\mathit{C}}_w+{\mathit{C}}_s{)}^{\frac{1}{2}}.$

From Equation (7) we know that ${\mathit{C}}_w+{\mathit{C}}_s$ under $H_1$, so $\tilde{\mathit{y}}=({\mathit{C}}_w+{\mathit{C}}_s{)}^{-\frac{1}{2}}\mathit{y}$ is distributed as $\mathcal{C}\mathcal{N}(0,I_M)$. Based on these, we can define the eigendecomposition of $\mathit{W}$ as where $\mathit{G}=diag\{f\left(\mathit{g}\right)+{\sigma }_{\theta }^{2}f{\left(\mathit{g}\right)}^2,0\dots 0\}$ and Equation (13) becomes where (a) is that the unitary transformation $\mathit{U}$ does not change the distribution of $\tilde{\mathit{y}}$, (b) holds since ${\tilde{\mathit{y}}}^H\mathit{G}\tilde{\mathit{y}}$ has a scaled Chi-square distribution with two degrees of freedom.

Similarly, the probability that a wireless signal does not exist and is detected $P_{FA}$ is

According to the Neyman–Pearson rule, by adjusting the decision threshold $\widehat{\gamma }$, the detection probability $P_D$ is maximized under a given false alarm probability $P_{FA}=\epsilon$. According to Equation (17), we have $\widehat{\gamma }=-ln\left(\epsilon \right){\sigma }_{\theta }^{2}f\left(\mathit{g}\right)$. Substitute $\widehat{\gamma }$ into Equation (16), we have

In this section, we using computer simulation to verify the correctness and effectiveness of the theoretical results obtained in previous sections, the simulation parameters which main reference [31] were set as follows: First, we consider $N=10$ sensors are evenly dispersed at a random distance from the M antennas of FC. The distances $d_i$ were uniformly distributed over [2, 20], in order to compare the effect of multi-antenna sensor and single-antenna sensor on detection performance in the same environment, we weaken the effect of distance on detection performance. For better illustration, a set of uniformly distributed random distance vectors is selected for subsequent simulation. The set of random distance vectors is [17.5024, 3.3541, 19.4117, 6.4284, 5.9324, 14.0173, 8.3763, 8.6594,19.7629, 11.0626]. The sum of the reported power of all the sensors is $P=400$ mW, the signal variance ${\sigma }_{\theta }^2=1$, and the noise variance ${\sigma }_n^2=0.3$. The false alarm probability $P_{FA}=\epsilon =0.05$ is temporarily given, which means that there is 5% probability that the absence of the signal will be falsely reported as the presence of the signal. In addition, the path loss index factor $\alpha =2$, signal variance ${\sigma }_{\theta }^2=0.05$, the measurement noise ${\sigma }_{v,ik}^2$ is uniformly distributed in the interval [0.25, 0.5] at the i-th sensor.

The Figure 2 depicts the relationship between the detection probability ($P_D$) and the false positive probability ($P_{FA}$) of the detector when the number of sensor antennas in the multi-antenna sensor network is $K=1,2,4$. It can be observed that when the number of antennas in the fusion center is fixed, the detection performance of the detector can be improved by using a multi-antenna sensor. When the sensor antenna is determined, the use of different antennas at the FC will also have an impact on the detection performance, with the number of antennas increases, the detection performance will be better, eventually reaching a performance limit as shown in Figure 3, there is still a certain gap between the theoretical value of the upper bound distance, which is caused by noise and actual equipment defects.

In Figure 4, the effect of the number of antennas of the multi-antenna sensor, the number of fusion center antennas and the total reported power on the detection performance are more clearly shown in the multi-antenna sensor network. The simulation results show that the use of multi-antenna sensors will bring better detection performance. We can see from the simulation results that as the number of FC antennas increases, when M reaches a certain antenna number (such as 200-250), the detection performance of the proposed solution is very close to the performance ceiling. According to our analysis combined with Equation (22), with the increase of the number of antennas in the fusion center, the reception capability of the fusion center will be stronger, and the detection capability to overcome fading in the transmission channel will be greatly improved, but when it reaches a certain level, the detection performance depends on other factors of the receiver noise, i.e., the certain antenna M is dependent on the number of sensor antennas and other variables(e.g., ${\sigma }_{\theta }^2$, ${\sigma }_{v,ik}^2$). In addition, with the infinite increase of the number of fusion center antennas or the unlimited power constraints, the detection performance of the system will reach the upper limit. As deduced from Equation (22), if the number of antennas and power at the FC are infinite, it is not required for power optimization, but in practice, the number of antennas cannot be too large and the power is limited. Therefore, it is important to consider optimizing power. This also gives us a thought that in the use of multi-antenna sensors, in order to obtain good detection performance, we find a trade-off relationship between the number of antennas M and the reported total power of P. In other words, we can use more antennas at limited power to compensate for the loss performance and vice versa.

Figure 5 depicts the proposed algorithm compared with the average power algorithm at M = 250. It can be seen that both algorithms can achieve high detection performance when the reported power P is large. When P is small through power optimization, our proposed solution can still maintain high detection performance.

Figure 6 shows that the proposed ADMM algorithm compares with the equal power allocation (EPA) algorithm when the power is 400 mW. It can be observed that when the antennas number M at FC is large, both ADMM and EPA algorithms can achieve high detection performance, and the proposed algorithm can better approach its performance ceiling. When M small, through the power optimization, our proposed solution can still maintain high detection performance. Furthermore, from the power allocation results of different sensors and antennas under the sum of reporting power in Table 1, our algorithm has different power due to different sensor positions and different noise impacts compared to the equal power allocation algorithm. Such as sensor 2, with the number of antennas increases, the required power is smaller. In addition, we can see that when the number of sensor antennas is fixed, the power allocation results of the different sensor antennas are almost the same. According to the analysis, because different antennas of the same sensor are almost the same distance from the FC, and the antenna noise has a slight effect on the power allocation result.

Figure 7 shows that the convergence performance of the ADMM iterative algorithm using different antenna sensors. Also, we can see that the proposed algorithm reaches a relatively stable state after 30 iterations. In that, a few tens of iterations will produce acceptable results of practical use. The timeliness of the algorithm is important for cooperative perception. This simulation result further validates the convergence speed of the proposed algorithm. For large distributed sensor networks, the faster the convergence speed of the algorithm, the faster the detection speed, which can meet the needs of quick detection.

In the context of the era of Internet of Things, the perception of the environment is particularly important for the use of detection networks. In this paper, we investigate the role of multi-antenna sensors in improving network perception performance. According to Neyman–Pearson criterion and maximum likelihood test, we derive a closed-loop expression for the detection probability of the best detector. To overcome the power limitation of the sensor, the sensors’ power was optimized by the ADMM. Finally, it is verified by simulation results that the multi-antenna sensor has better detection performance than single-antenna sensor. The optimal detection probability can be obtained with the least energy consumption by optimizing the power. It will be interesting and important to investigate the role of multi-antenna diversity in the detection network and the analysis of detection performance under irrational channel conditions and the presence of device hardware impairments in future work.